Determining positions of transducers for receiving and/or transmitting wave signals

ABSTRACT

In a method for determining positions {p i } i=1, . . . , N  of transducers {A i } i=1, . . . , N  of an apparatus, the transducers are assumed to be configured for receiving wave signals from and/or transmitting wave signals to one or more regions {R m } m=1, . . . , M  of interest in an n-dimensional space, with n=2 or 3. An n-dimensional spatial filter function {circumflex over (ω)} e (r) is determined, which matches projections {P m } m=1, . . . , M  of the one or more regions {R m } m=1 . . . , M  of interest onto an n−1-dimensional sphere centered on the apparatus. Then, a density function ƒ b (p) is obtained, based on a Fourier transform ω(p) of the determined spatial filter function {circumflex over (ω)} e (r). Finally, a position p i  is determined, within said n-dimensional space, for each of N transducers, based on the obtained density function ƒ b (p) and a prescribed number N of the transducers. The invention is further directed to related devices, apparatuses and systems, as well as computer program products.

BACKGROUND

The invention relates in general to methods for determining positions oftransducers (e.g., antennas) configured for receiving and/ortransmitting wave signals, as well as related devices and systems. Inparticular, it is directed to methods for designing phased-arraysaccording to a specified, target beamshape. It also relates tobeamforming techniques, i.e., methods for beamforming signals orcomputing beamformed signals, by explicitly targeting designed regionsof interest.

Beamforming combines networks of antennas or sensors coherently so as toachieve a beamshape with desirable properties such as high directivity,reduced side-lobes, and improved signal-to-noise ratio (SNR).

Beamforming is often thought of as steering the array to focus on aparticular point. This conception, however, has a number of drawbacks.It cannot adjust for small errors in direction of interest estimates,nor cope well with moving devices. Additionally, often a region orregions is what is desired, e.g., to scan the sky in radio astronomy oran organ in ultrasound.

The so-called flexibeam framework [P. Hurley and M. Simeoni,“Beamforming towards regions of interest for multi-site mobilenetworks,” in International Zurich Seminar on Communications, 2016, p.94] was developed to determine, in a data independent fashion,beamforming weights so as to target a general spatial region. Theframework takes as input fixed antenna locations, and approximates thetarget beamshapes.

SUMMARY

According to a first aspect, the present invention is embodied as amethod for determining positions {p_(i)}_(i=1, . . . , N) of transducers{A_(i)}_(i=1, . . . , N) of an apparatus. The transducers are assumed tobe configured for receiving wave signals from and/or transmitting wavesignals to one or more regions {R_(m)}_(m=1, . . . , M) of interest inan n-dimensional space, with n=2 or 3. The method relies on ann-dimensional spatial filter function {circumflex over (ω)}_(e)(r),which is determined so as to match projections {P_(m)}_(m=1, . . . , M)of the one or more regions {R_(m)}_(m=1, . . . , M) of interest onto ann−1-dimensional sphere centered on the apparatus. Then, a densityfunction ƒ_(b)(p) is obtained, based on a Fourier transform ω(p) of thedetermined spatial filter function {circumflex over (ω)}_(e)(r).Finally, a position p_(i) is determined, within said n-dimensionalspace, for each of N transducers, based on the obtained density functionƒ_(b)(p) and a prescribed number N of the transducers.

According to another aspect, the invention is embodied as a systemcomprising an apparatus with a set of N transducers{A_(i)}_(i=1, . . . , N). The transducers are configured, in theapparatus, for receiving wave signals from and/or transmitting wavesignals to one or more regions {R_(m)}_(m=1, . . . , M) of interest inan n-dimensional space, with n=2 or 3. Moreover, and consistently withthe above method, the transducers are arranged at positions{p_(i)}_(i=1, . . . , N) within said n-dimensional space, wherein saidpositions {p_(i)}_(i=1, . . . , N) corresponds to arguments of values{ƒ_(b)(p_(i))}_(i=1, . . . , N) of a density function ƒ_(b)(p) obtainedbased on the Fourier transform ω(p) of the n-dimensional spatial filterfunction {circumflex over (ω)}_(e)(r). As said above, the latter isdetermined so as to match projections {P_(m)}_(m=1, . . . , M) of theone or more regions {R_(m)}_(m=1, . . . , M) of interest onto then−1-dimensional sphere, centered on the apparatus.

Finally, and according to a further aspect, the invention can beembodied as a computer program product for determining positions of suchtransducers. The computer program product comprises a computer readablestorage medium having program instructions embodied therewith, whereinthe program instructions are executable by a computerized system tocause the latter to implement steps according to the present methods.

Devices, apparatuses, systems, methods and computer program productsembodying the present invention will now be described, by way ofnon-limiting examples, and in reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 represents a plan of a living room overlaid with a preferencefunction, which identifies regions of interest (desk R₁, sofa R₂ andkitchen R₃) in the living room as a density plot. Darker regionsrepresent regions of higher interest, where users most often need toreceive and transmit signal. A router (circle) is located at theupper-right corner; a plot is overlaid that schematically represents atarget angular filter function {circumflex over (ω)}(θ), suited forcovering the regions of interests, as involved in embodiments;

FIGS. 2A-2B depict the same target angular function {circumflex over(ω)}(θ) on a unit circle, as well as a corresponding spatial filterfunction {circumflex over (ω)}_(e)(r), which is a bi-dimensionalextension of {circumflex over (ω)}(θ), as in embodiments. The function{circumflex over (ω)}_(e)(r) is represented as a density plot,contrasted for the sake of depiction, which shows three main peaks thatmatch {circumflex over (ω)}(θ) on the unit circle. In more detail:

In FIG. 2A, selected portions are shown as dotted, white doubles arrowson the unit circle, which portions correspond to highest values ofprojections P₁, P₂, P₃ of respective regions R₁, R₂, R₃ on the unitcircle. Regions R₁, R₂, R₃ are located farther away from the center ofthe circle;

FIG. 2B further shows the resulting beamforming filter b_(N)(θ), i.e., abeamshape obtained according to beamforming weights ω(p_(i)), which takepositions {p_(i)}_(i=1, . . . , N) of the transducers (antennas) asarguments. These positions have been optimized thanks to a densityfunction ƒ_(b)(p), based on the Fourier transform ω(p) of {circumflexover (ω)}_(e)(r), taking a prescribed number of antennas as input,according to embodiments. The final beamshape b_(N)(θ) is very close tothe target angular filter;

FIG. 3 depicts the angular filter {circumflex over (ω)}(θ) otherwiseunwrapped, where it is plotted over the interval [π/2; π]. As furtherseen in FIG. 3, the angular filter {circumflex over (ω)}(θ) can beapproximated as a combination of functions (here Gaussians) and so doesthe corresponding spatial filter {circumflex over (ω)}_(e)(r) (not shownin this figure), in embodiments;

FIGS. 4A-4C, 5A-5C, 6A-6C, and 7A-7C (collectively FIGS. 4-7) showexamples of beamshapes b_(N)(θ), and transducer layouts{p_(i)}_(i=1, . . . , N) as obtained from associated density functionƒ_(b)(p), using various intermediate (basis) functions to obtain aspatial filter {circumflex over (ω)}_(e)(r) that, in this example, has asingle focus direction r₀. The angular filter {circumflex over (ω)}(θ)is determined based on the circular sector enclosed by two (dotted)radii. FIGS. 4-7 show results obtained using: a so-called “sombrero”function; a radial Laplace function; a ball-indicator function; and abi-dimensional Gaussian function, respectively. The LHS figures (4A, 5A,6A, 7A) depict, each, the target angular filter {circumflex over (ω)}(θ)and its spatially extended counterpart {circumflex over (ω)}_(e)(r),whereas the RHS figures (4C, 5C, 6C, 7C) show the resulting beamshapesb_(N)(θ). The middle figures (4B, 5B, 6B, 7B) show the antenna arraylayouts {p_(i)}_(i=1, . . . , N), overlaid on a density plot of thebeamforming density function ƒ_(b)(P);

FIG. 8 is a 3D view of a router, comprising 27 antennas, arranged in therouter according to a layout obtained according to embodiments; and

FIG. 9 schematically represents an array of ultrasound transducers,arranged according to a layout obtained according to embodiments, andtransmitting original sound waves (plain lines) and receiving waves(dashed lines) reflected by a sample;

FIG. 10 is a flowchart that illustrates high-level steps of a method fordetermining positions of transducers of an apparatus, according toembodiments; and

FIG. 11 shows a computerized system suited for implementing one or moremethod steps as in embodiments of the invention.

The accompanying drawings show simplified representations of devices andcomponents (e.g., transducers) thereof, as involved in embodiments.Similar or functionally similar elements in the figures have beenallocated the same numeral references, unless otherwise indicated.Density plots represented in the figures have been purposely applied ahigh contrast, for the sake of depiction.

DETAILED DESCRIPTION

The present Inventors noticed that the existing beamforming techniques,although powerful, gave no hints on where to place the transducers. Forinstance, The flexibeam framework evoked in the background section takesas input fixed antenna locations to approximate the target beamshapes.Therefore, the Inventors took up the challenge to provide solutions tothe problem of transducer positioning. They have accordingly achievedmethods for designing transducer layouts based on given, specifiedtarget regions, wherein transducer locations are obtained from aprobabilistic density function. Such methods are all the more suited forapplications to phased-arrays, taking as input a given, targetbeamshape.

The following description is structured as follows. First, generalembodiments and high-level variants are described (sect. 1). The nextsections (sect. 2-4) describe specific embodiments, examples ofapplications, and technical implementation details, respectively.

1. General Embodiments and High-Level Variants

In reference to FIGS. 2-10, an aspect of the invention is firstdescribed, which concerns methods for determining positions{p_(i)}_(i=1, . . . , N) of transducers {A_(i)}_(i=1, . . . , N) of anapparatus 10, 10 a.

Generally speaking, the transducers are devices configured for receivingwave signals from and/or transmitting wave signals to one or moreregions {R_(m)}_(m=1, . . . , M) of interest. The devices{A_(i)}_(i=1, . . . , N) may be receivers, transmitters or transceivers.In all cases, such devices convert a signal in one form of energy to asignal in another form of energy. In particular, the transducers mightbe bidirectional, e.g., like antennae, which can convert electricalsignals to or from propagating electromagnetic waves, and ultrasoundcoils, which convert electrical signals into/from ultrasounds. Ingeneral, the signals received by and/or transmitted from the transducerswill be waves, e.g., electromagnetic waves or mechanical waves, likesound waves.

The regions {R_(m)}_(m=1, . . . , M) are defined in an n-dimensionalspace (or n-space), where n=2 or 3. Each of the transducers need bepositioned in the n-space. That is, the position p; of each device{A_(i)}_(i=1, . . . , N) must be determined in

^(n), i.e., in the three dimensional space (if n=3) or a plane (if n=2).

To that aim, the present methods first determine an n-dimensionalspatial filter function {circumflex over (ω)}_(e)(r), as, e.g., in stepsS12-S15 of FIG. 10. The spatial filter function {circumflex over(ω)}_(e)(r) (also called spatial filter, or extended filter function inthe following) is determined as a function defined in the n-space. Thisfunction need to match projections {P_(m)}_(m=1, . . . , M) of regions{R_(m)}_(m=1, . . . , M) of interest onto an n−1-dimensional sphere (orn−1-sphere, for short). One assumes there is at least one region ofinterest for the present purpose.

The n−1-sphere is assumed to be centered on the apparatus 10, 10 a. Forinstance, this sphere may, eventually, be centered on the barycenter ofthe positions {p_(i)}_(i=1, . . . , N) of the transducers (yet to bedetermined). The above concepts of n-space, n−1-sphere, regions ofinterest and projections are discussed below in detail.

Next, a density function ƒ_(b)(p) is obtained S16, based on a Fouriertransform ω(p) of the previously determined spatial filter function{circumflex over (ω)}_(e)(r). As it may be realized, the Fouriertransform of {circumflex over (ω)}_(e)(r) may indeed be interpreted as adistribution function, for reason that will become apparent later. Thedensity function ƒ_(b)(p) may for instance be obtained S16 as a functionproportional to the absolute value of the Fourier transform ω(p) of theS15 spatial filter function {circumflex over (ω)}_(e)(r). Furthermore,the density function ƒ_(b)(p) is preferably suitably normalized.Additional mathematical details are given in sect. 2.

Finally, assuming a prescribed number N of transducers, a position p_(i)can be determined S17-S18, within the n-space, for each of the Ntransducers, based on the obtained density function ƒ_(b)(p), e.g., bysuitably sampling the latter. A key point to enable this procedure is tomake sure that {circumflex over (ω)}_(e)(r) is defined in the n-space,i.e., that {circumflex over (ω)}_(e)(r) take values for r in then-space, and not in the n−1-space.

Consider for instance an array of N receiving transducers{A_(i)}_(i=1, . . . , N): application of the present methods allowpositions {p_(i)}_(i=1, . . . , N) of the transducers to be determined,given specified regions of interest, where {p₁, . . . , p_(N)}∈

^(n). When n=3, the positions determined lie in the 3D space, and whenn=2, they lie on a plane. That the space the transducers lie in isallowed to vary permits different embodiments, leading to differentspatial arrangements of the transducers, depending on the application athand.

The present methods can advantageously be applied to phased arrays, forwhich the optimization and design of the layout of the transducers willcritically impact the performance. For example, the beamshape of anantenna array depends both on the geometrical (layout) and electrical(gain and phase at each antenna) properties of the array. While thegeometrical (layout) properties of the array is usually considered as afixed input, i.e., a constraint, the present approach makes it possibleto design a layout for the transducers according to specified targets,located in regions of interest. Projecting such regions onto then−1-sphere (centered on the apparatus) allows a spatial filter to bedesigned and, in turn, a density function to be achieved, from which thepositions of each transducer are derived. In fine, an apparatus can bedesigned and built, whose transducers are arranged according topositions determined thanks to present methods.

During a second phase (i.e., after having determined the positions{p_(i)}_(i=1, . . . , N) of the transducers), apparatuses designed andbuilt thanks to the present methods may be used for beamforming signals(for transmission purposes) or compute beamformed signals (for receptionpurposes). Such signals are to be respectively transmitted or receivednon-uniformly in a 2- or 3-dimensional space, i.e., according to theparticular regions of interest. To that aim, delays and gains maysuitably be introduced in the signals, by weighting time-seriesaccording to beamforming weights ω(p_(i)).

That is, the beamforming weights ω(p_(i)) are easily be computed thanksto the determined positions and the Fourier transform ω(p) of{circumflex over (ω)}_(e)(r), by evaluating ω(p) for each positionp_(i). Then, delays and gains are introduced, by weighting time-seriesfrom or for the devices {A_(i)}_(i=1, . . . , N), using beamformingweights ω(p_(i)). Namely, each time series received from or to betransmitted to a distinct one A_(i) of the devices is weighted by abeamforming weight ω(p_(i)) as determined for said distinct one A_(i) ofthe devices.

The core principle remains the same, that signals be transmitted orreceived. For example, in a transmission case, signals are beamformed byintroducing delays and gains as described above, i.e., by weightingtime-series which can then be forwarded to respective transmitters(e.g., antennas of a Wifi router), for subsequent transmission to finalreceivers (e.g., end-users or signals consumers, such as laptops,smartphones, etc.). In reception, signals are sent by initialtransmitters (e.g., the same end-users) to receivers (e.g., the sameantennas).

Interestingly, the present approach makes it possible to focus onspecific regions of interest. Aiming for regions (or areas) rather thanpoints minimizes the required update rate and reduces the requirementfor communication. The techniques used herein allow wider ranges ofdirections to be covered than with matched beamforming techniques. Thatis, for matched beamforming to cover a same range as with the presentapproach (all things being otherwise equal), one would have to steermatched beamforming toward many directions within that range. Moreover,and as illustrated in the accompanying drawings, the present solutiontypically gives rise to less aliasing artifacts (e.g., side-lobes) inthe beamshapes.

The present solution can advantageously be used in various applications,including mobile telecommunication networks, Wifi, radio astronomy andultrasound apparatuses, with, the devices {A_(i)}_(i=1, . . . , N) beingreceivers, transmitters or transceivers.

In embodiments, the spatial filter {circumflex over (ω)}_(e)(r) isobtained based on a preference function (step S12, FIG. 10), whichidentifies the regions of interest {R_(m)}_(m=1, . . . , M), in

^(n). Such regions can be regarded as preferential regions, from whereor to where signals need mostly be sent, e.g., because of a non-uniformdistribution, or concentration of the end-users (i.e., the targetsand/or the initial transmitters). Accordingly, signals can be sent withhigher intensity to or from such regions.

As the regions of interest may evolve, the preference function maychange accordingly, thereby triggering a re-configuration S10-S11,leading to updated beamforming weights ω(p_(i)). That is, the system 1,10, 10 a is preferably reconfigurable, i.e., the layout of thetransducers may be dynamically changed. In that case, the configurationof the system may be monitored S10-S11. If the configuration of thetargets happens to change S11, the updated configuration may be capturedin an updated preference function, which gets loaded into the system andnew positions can be calculated according to principles discussed above.In variants, even if the system is not reconfigurable, the beamshapesmay nevertheless be reconfigured base on the (non-reconfigurable layoutof antennas). Obviously, such a system is less performant, though moresimple. In all cases, the present methods allow some versatility,compared to prior art approaches.

Having identified S12 one or more regions thanks to the preferencefunction, the regions can then be projected S13 onto the n−1-sphere.This way, an angular filter function, also referred to as “targetfilter” in this document, can be obtained S14, based on the projections.The angular filter function is denoted by {circumflex over (ω)}(θ) inFIGS. 2-7, in which n=2, for simplicity. For applications in the 3Dspace, the angular filter function may need be defined as {circumflexover (ω)}(θ, φ) or, more generally, as {circumflex over (ω)}(r). Thetarget filter is sometimes noted {circumflex over (ω)}(θ, . . . ) in thefollowing, to indicate it may take values on a unit circle or a unitsphere.

Note that:

-   -   A 1-sphere is a circle and is the boundary of a disk (also        called a 2-ball); and    -   A 2-sphere is an ordinary sphere in 3-dimensional Euclidean        space, and is the boundary of an ordinary ball.

Projecting one or more regions of interest onto an n−1-sphere gives riseto one or more “surface portions”, respectively. Thus, the projections{P_(m)}_(m=1, . . . , M) correspond to respective “surface portions”,where a surface portion is an arc of a circle if n=2 or a surfaceportion of an ordinary sphere if n=3.

For completeness, projections on an n−1-sphere correspond ton−1-dimensional angles that the regions subtend at the center of then−1-sphere. Such angles are planar angles when n=2 or solid angles whenn=3. The projected regions correspond to projections (or equivalentlysurface portions) as the set of transducers “see” them from within then−1-sphere, assuming that this sphere is large enough to contain each ofthe transducers. However, the actual size of this sphere has littleimportance, for reasons that will become apparent later. Thus, ann−1-sphere as considered herein is typically a unit sphere (i.e., anobject defined in the n−1-space).

The angular filter function {circumflex over (ω)}(θ, . . . ) can be seenas an optional intermediate between the projections of the regions onthe n−1-sphere and the extended filter {circumflex over (ω)}_(e)(r). Theangular filter {circumflex over (ω)}(θ, . . . ) is constructed based onsuch projections, so as to reflect values taken by the projections onthe n−1-sphere. As per its construction, {circumflex over (ω)}(θ, . . .) is defined on the sole n−1-sphere. Yet, the spatial filter function{circumflex over (ω)}_(e)(r) can easily be obtained by extending theangular filter function {circumflex over (ω)}(θ, . . . ) to

^(n).

As a result, {circumflex over (ω)}_(e)(r) as obtained at step S15extends (i.e., takes values) in

^(n) while the projections are n−1-dimensional objects. That is, thefunction {circumflex over (ω)}_(e)(r) “extends” the projections{P_(m)}_(m=1, . . . , M) over

^(n) and, as such, are “thicker” objects, defined in a space of largerdimension than the projections.

The spatial filter function {circumflex over (ω)}_(e)(r) plays animportant role. It need be designed so as to match the projections ofthe regions onto the n−1-sphere, i.e., so as to approximate theseprojections on the n−1-sphere, it being noted that {circumflex over(ω)}_(e)(r) is not only defined on this sphere but also in the n-space.For instance, the function {circumflex over (ω)}_(e)(r) may beconstrained to take the same values as the angular filter {circumflexover (ω)}(θ, . . . ) on the n−1-sphere. The function {circumflex over(ω)}_(e)(r) will otherwise typically exhibit peaks that coincide withmaxima of such projections, on the n−1-sphere. When considering suchpeaks as objects extending in

^(n) (which they are, as opposed to the angular filter that isrestricted to the n−1-sphere), then such peaks typically circumscribe ortightly encompass the projections on the n−1-sphere, as seen in FIG. 2A.

In other words, {circumflex over (ω)}_(e)(r) “extends” the angularfunction {circumflex over (ω)}(θ, . . . ) in

^(n), notably radially (i.e., perpendicular to the n−1-sphere) and is a“thicker” object than {circumflex over (ω)}(θ, . . . ). As presentinventors have realized, {circumflex over (ω)}_(e)(r) indeed need be“thicker” than {circumflex over (ω)}(θ, . . . ) because an objectrestricted to the sole n−1-sphere (like (θ, . . . )) would make itimpossible to recover a beamforming function or, even, to obtain ameaningful layout for the transducers.

As explained below in detail, one or more peaks of {circumflex over(ω)}_(e)(r) may be used to match projections of the regions of interest.Here, the terminology “peak” is to be understood broadly: a peakreflects values taken by {circumflex over (ω)}_(e)(r) in a given portionof the 2- or 3-space, where {circumflex over (ω)}_(e)(r) takes highervalues than around this portion. A peak may even be a 2-dimensionalrectangular (or boxcar) function, i.e., involve an infinite number oflocal maxima.

Preferably, {circumflex over (ω)}_(e)(r) is a L2-function, i.e., it issquare-integrable over

^(n) and is finite, which implies that |{circumflex over (ω)}_(e)(r)|→0as |r|−∞. As a L2-function, {circumflex over (ω)}_(e)(r) suitably behavefor it to potentially match projected regions. More preferably, thefunction {circumflex over (ω)}_(e)(r) is one that satisfies allDirichlet conditions, for it to be Fourier transformable. While{circumflex over (ω)}_(e)(r) may, in some cases, be always positive, ∀rin

^(n), it may, in other cases, be chosen to be partly negative. I.e.,negativity could be desirable in some cases, for example when using thepresent methods to expand an incoming signal in a desired basis, whoseelements may be negative. More generally, the function {circumflex over(ω)}_(e)(r) may be complex.

The positions of the transducers are preferably determined S17-S18 bysampling S17 values {ƒ_(b,j)}_(j=1, . . . , N) of the density functionƒ_(b)(p) obtained at step S16. From the sampled values{ƒ_(b,i)}_(i=1, . . . , N), one may then identify S18 correspondingarguments p_(i) corresponding to the sampled values, i.e., such thatƒ_(b)(p_(i))=ƒ_(b,i) ∀i. The final positions of the N transducers arethen determined based on the identified arguments{p_(i)}_(i=1, . . . , N). note that the final positions of the Ntransducers may simply be equated to the identified arguments{p_(i)}_(i=1, . . . , N). Yet, additional (e.g., design) constraints mayneed be taken into account, which may lead to small deviation from theideal positions {p_(i)}_(i=1, . . . , N).

The geometry of the transducers can for instance be constructed from thequantile function of the density function ƒ_(b)(p). By appropriatelysampling this function, the resulting beamshape will closely resemblethe target.

Preferably, the spatial filter function {circumflex over (ω)}_(e)(r) isdetermined S15 as a function having a defined, analytical Fouriertransform, such that the density function ƒ_(b)(p) can next be obtainedS16 based on this analytical Fourier transform. E.g., the spatial filterfunction {circumflex over (ω)}_(e)(r) may be constructed based on known,well-defined analytical functions, for which analytical Fouriertransforms are known, such that no explicit Fourier transformation needbe involved. Thus, the function ω(p) can be obtained based on readilyavailable, analytical Fourier transforms. All the more, no numericalFourier transformation is involved in that case, which avoids numericalinstabilities as typically caused by numerical transforms. In lesspreferred variants though, the present methods may also be implementedthanks to numerical transforms.

Using filter functions that have defined, analytical Fourier transformsallows a procedure that is numerically stable, computationally efficientand flexible, which, in turn, offers analysts a powerful tool fordesigning beamshapes in a much more controlled manner. That thetransducers layout is optimized allows the quality of the subsequentbeamshapes to be substantially improved. In that respect, and as seen inFIGS. 4-7, layouts as obtained thanks to embodiments of this inventionmay lead to beamshapes never obtained before and never thoughtobtainable.

Depending on the complexity desired for the spatial filter function{circumflex over (ω)}_(e)(r), one or more auxiliary functions may beinvolved in its construction, where each auxiliary function has adefined, analytical Fourier transform. Namely, in embodiments, thefunction {circumflex over (ω)}_(e)(r) is determined S15 as a sum ofauxiliary functions that have, each, a defined, analytical Fouriertransform. In that case, the density function ƒ_(b)(p) is obtained S16based on a sum of analytical Fourier transforms that correspond, each,to a respective one of the auxiliary functions.

Using a sum of auxiliary functions makes it possible to closely fit avariety of target functions, even functions having cusps or edges. Thespatial filter function {circumflex over (ω)}_(e)(r) may for instance bedetermined as a sum of Gaussian functions. This is advantageous ascontractions of even a few Gaussians can be used to fit a wide range oftarget functions, including functions having cusps, and, this, to areasonable approximation. Furthermore, Gaussian functions allow for aneasy representation in each of the direct space and the transformedspace. A Gaussian function that is localized in one space, will bediffuse in the transformed space. However, as the transform of aGaussian is still a Gaussian, the transformed function is stillwell-behaved; this not only eases the transforms but also avoidsspurious effects when computing the weight coefficients and simplifiesthe computation of such coefficients. In other words, well-behavedcoefficients can be obtained, which lead to clean beams.

In variants, ball indicator functions may be used. Other examples offunctions can be contemplated, as discussed in sect. 2. The auxiliaryfunctions used need not form a complete basis. On the contrary, acombination of different types of functions may possibly be used (in aweighted sum of such functions). E.g., one may simultaneously use ballindicator functions, Gaussian functions, Slater functions, etc., wheresome of the Gaussian/Slater functions may, if necessary, beparameterized so as to approach a Dirac distributions. This, in turn,allow more complex filters to be approximated.

In still other variants, however, one may rely on auxiliary functions,which do not satisfy Dirichlet conditions, but which locally behavewell, e.g., take the same (or almost the same) values as known,transformable functions. The function {circumflex over (ω)}_(e)(r) mayfor instance be written as a weighted (finite) sum of sine and/or cosinefunctions, fitting to a target angular filter {circumflex over (ω)}(θ),see explanations below, on the n−1-sphere, while the transform of{circumflex over (ω)}_(e)(r) is taken as an integral over a finitedomain (which again yields analytical formula). More generally, the oneskilled in the art will appreciate that many types of auxiliaryfunctions may be contemplated.

The auxiliary functions used to construct {circumflex over (ω)}_(e)(r)may further be extrapolated from the auxiliary functions used toconstruct the target angular filter, as illustrated in FIG. 3. Thispoint is discussed in more detail in sect. 2.

Where {circumflex over (ω)}_(e)(r) is taken as a sum of auxiliaryfunctions, the density function ƒ_(b)(p) is preferably obtained as theabsolute value of the sum of the sum of the Fourier transforms of eachindividual auxiliary function, for reasons that will become apparent insect. 2.

In simple embodiments, M auxiliary functions are used, which correspondsto the number of regions {R_(m)}_(m=1, . . . , M) of interest. I.e., theauxiliary functions are chosen so as to match projections{P_(m)}_(m=1, . . . , M) of the regions {R_(m)}_(m=1, . . . , M) ontothe n−1-sphere, as illustrated in FIG. 2A.

More generally, {circumflex over (ω)}_(e)(r) may be devised so as toexhibit, on the n−1-sphere, maxima that match the maxima of theprojections {P_(m)}_(m=1, . . . , M), so as to suitably reflect theangular distribution of the regions {R_(m)}_(m=1, . . . , M) onto then−1-sphere. Now, and depending on the auxiliary functions used,{circumflex over (ω)}_(e)(r) may have more (or less) local maxima thanthe projections {P_(m)}_(m=1, . . . , M) of the regions on then−1-sphere.

In simple cases, the angular filter {circumflex over (ω)}(θ, . . . ) maytake on the value 1 on certain “arcs”, i.e., portions of the n−1-sphere(corresponding to projected regions), and 0 outside, e.g., if regionsare initially identified in a binary way rather than with a density map.In such cases too, the auxiliary functions may be suitably chosen so asfor {circumflex over (ω)}_(e)(r) to faithfully reflect the angularfilter on the n−1-sphere. For instance, the spatial filter {circumflexover (ω)}_(e)(r) may, in that case, be designed so as to have, on acircle (n=2) or on a sphere (n=3), one or more maxima (or a constantvalue) within each of the one or more arcs or surface portionscorresponding to the projections {P_(m)}_(m=1, . . . , M) of the regionsof interest.

Referring now more specifically to FIGS. 1, 8 and 9: according toanother aspect, the invention can be embodied as a system 1 comprisingan apparatus 10, 10 a such as depicted in FIGS. 8 and 9. This apparatuscomprises a set of N transducers {A_(i)}_(i=1, . . . , N), which areconfigured, in the apparatus 10, 10 a, for receiving wave signals fromand/or transmitting wave signals to one or more regions of interest inthe n-space (with, again, n=2 or 3).

In addition, the transducers are arranged at specific positions{p_(i)}_(i=1, . . . , N), i.e., as obtained thanks to the presentmethods. For example, the positions of the transducers may correspond toarguments of values of the density function ƒ_(b)(p), the latterobtained based on the Fourier transform ω(p) of {circumflex over(ω)}_(e)(r), as described above. The resulting layout is atypical, asseen in FIGS. 4C to 7C. It may, at first sight, evoke a randomdistribution and clearly departs from usual layouts of transducers,which are typically arranged according to a regular array, having givensymmetries.

As evoked earlier, the system 1 is optionally reconfigurable. To thataim, the transducers may be movably mounted in the apparatus 10, 10 a,or, at least, the transducers are re-positionable (as typically is thecase during fabrication). A reconfigurable system need be suitablyequipped (with movable parts, stages, etc.) for (re-)positioning thetransducers according to positions determined based on given targets.

The present systems will typically comprise computerized modules,configured to perform the following tasks (one or more modules may beinvolved, having one or more functions each):

-   -   First, such modules need be designed to determine a spatial        filter function, so as for the latter to match projections of        the regions of interest onto the n−1-sphere (centered on the        apparatus 10, 10 a). To that aim, the modules may rely on a        preference function, compute the angular filter and, based on        it, determine a suitable extended filter. Additional modules may        possibly monitor the configuration, and trigger a recomputation        of {circumflex over (ω)}_(e)(r) when the configuration changes.    -   Second, the modules will be adapted to determine the density        function ƒ_(b)(p), by Fourier transforming the previously        determined spatial filter function, as explained earlier. In        fact, where analytical Fourier transforms are available, such        modules will simply compute ƒ_(b)(p) based on the known        analytical transforms; and    -   Third, the modules need be configured to determine positions of        the transducers, based on the density function ƒ_(b)(p), e.g.,        by suitably sampling the latter.

In addition, the present systems may comprise:

-   -   A digital beamforming module, configured for computing a        beamformed signal, based on a summation of time-series received        by receivers (or transceivers); and/or    -   An analog beamforming module, configured for beamforming        signals, by weighting time-series to be transmitted by        transmitters (or transceivers). In this case, the beamforming        module can indeed be regarded as an analog module, inasmuch as        delays and gains are introduced in the signals to be        transmitted, via the weighting scheme.

Such a system may notably be embodied as: a transmitter apparatus; areceiver; a transceiver (e.g., a wireless access point such as a Wifirouter, as in FIG. 8; a mobile telecommunication network, comprising oneor more of the afore-mentioned apparatuses; a large phased array ofantenna; or, still, as an ultrasonic sensing system, such as depicted inFIG. 9.

The system may comprises more than one set of transducers, the setspossibly including receivers, transmitters and/or transceivers, whichmay possibly form distinct modules, located in distinct regions ofspace. The transducers of at least one of these sets will be arranged atspecific positions {p_(i)}_(i=1, . . . , N), i.e., as obtained thanks tothe present methods. One or more of these sets of transducers may form“targets”, i.e., be located in said regions of interest (and yet formingpart of the system), whereas another one of the set may comprisetransducers arranged at said specific positions{p_(i)}_(i=1, . . . , N), so as to efficiently transmit signals to thetargets.

The above embodiments have been succinctly described in reference to theaccompanying drawings and may accommodate a number of variants. Severalcombinations of the above features may be contemplated. Examples aregiven in the next section.

2. Specific Embodiments—Implementation Details

This section describes a method that is useful for designingphased-arrays according to a given, analytically-specified, targetbeamshape. Building on the flexibeam framework, antenna locations aresampled from a probabilistic density function. Naturally scalable withthe number of antennas, it is also computationally efficient andnumerically stable, as it relies on analytical derivation.

It is further shown that, under mild conditions, the achieved beamshapesconverge uniformly to the target beamshapes as the number of antennasincreases. The technique is illustrated through a number of examples.For instance, by use of the Laplace filter, beams with extremely fastdecay away from the center of focus are achieved.

The procedure is numerically stable, computationally efficient andversatile, offering the analyst a very powerful tool for designingbeamshapes, as well as insight into how an array will scale with everincreasing numbers of antennas. Matched beamforming weights may, for agiven layout, achieve beamshapes targeting regions, rather than isolateddirections as commonly believed.

As the beamforming weights all have unit modulus, this translates into adelay, allowing existing hardware to be used to achieve a given beam,and has the advantage of treating noise uniformly across antennas. Incontrast, prior approaches like the so-called “flexibeam” techniquemodify the magnitude of the weights at the cost of non-uniform noisedistribution.

Additionally, the convergence analysis can be used to forecast thegrowth of future, large phased arrays such as the Square Kilometer Array(SKA).

For simplicity in explanation, this description merely considerscircularly symmetric filters and 2D beamforming. Extension to 3Dbeamforming readily follows.

2.1 Preliminaries

2.1.1 Beamforming

Consider an array of N receiving antennas, which, without loss ofgenerality, have unit gains and an omni-directional field of view. Inmathematical terms, “beamforming” translates into a linear combinationof the antenna signals x_(i)(t):

$\begin{matrix}{{{y(t)} = {\sum\limits_{i = 1}^{N}{w_{1}^{*}{x_{i}(t)}}}},} & (1)\end{matrix}$where ω_(i)∈

are beamforming coefficients.

Under the non-limiting assumptions of far-field and narrowband signals,

$\begin{matrix}{{y(t)} = {{\int_{0}^{2\pi}{{s\left( {t,\theta} \right)}{b^{*}(\theta)}d\;\theta}} + {\sum\limits_{i = 1}^{N}{w_{i}^{*}{{n_{i}(t)}.}}}}} & (2)\end{matrix}$

The function b(θ) is called the beamshape of the beamformed antennaarray and is defined as:b(θ)=Σ_(i=1) ^(N)ω_(i) e ^(j2πp) ^(i) ^(cos(θ-ϕ) ^(i) ⁾.It describes the sensitivity of the antenna array relative to variousdirections θ, and hence acts as an angular filter.

2.1.2 Beamforming with Flexibeam

The so-called flexibeam framework (see also P. Hurley and M. Simeoni,“Flexibeam: analytic spatial filtering by beamforming,” in InternationalConference on Acoustics, Speech and Signal Processing (ICASSP), IEEE,March 2016) permits to achieve a wide range of analytically-specifiedbeamshapes. In this section, some important notions and notationsborrowed from this framework are briefly recalled.

One innovation of Flexibeam was to consider a notional continuous fieldof antennas, with an associated beamforming function ω:

²→

, describing the gains and delays to be applied at each location (p,φ)∈(p, ϕ)∈

₊×[0,2π] so as to achieve an analytically specified target beamshape{circumflex over (ω)}: [0,2π]→

:{circumflex over (ω)}(θ)=∫₀ ^(∞)∫₀ ^(2π)ω(p,ϕ)e ^(j2πp cos(θ-ϕ))pdϕdp.  (3)

The beamforming function can be computed asω(p,ϕ)=∫₀ ^(∞)∫₀ ^(2π){circumflex over (ω)}_(e)(r,θ)e ^(−j2πrp cos(θ-ϕ))rdθdr  (4)where {circumflex over (ω)}_(e):

²→

is the extended filter introduced in the previous section, which can beconstrained such that {circumflex over (ω)}_(e)(1, θ)={circumflex over(ω)}(θ). For N given antenna locations, the corresponding beamformingweights are then obtained by sampling ω:

${w_{i} = \frac{\omega\left( {p_{i},\phi_{i}} \right)}{\beta}},{i = 1},{\ldots\mspace{14mu} N},$where β is a normalizing constant to avoid noise magnification. Theresulting beamshape then approximates the target beamshape, withaccuracy dependent on the specific layout under consideration.

2.2 “Flexarray”

In this section, a new framework is proposed, termed “flexarray”, togenerate random layouts so as to approximate analytically-specifiedtarget beamshapes. To do so, we leverage the flexibeam framework andconstruct a probability density function from the beamforming function.In the specific case of circularly symmetric extended filters, we linkthis density function to the Hankel transform of order zero of theextended filter. We finally show that the empirical beamshapes obtainedwhen sampling this density converge uniformly almost surely to thetarget beamshape as the number of antennas grows to infinity.

2.2.1 Beamforming with Circularly Symmetric Extended Filters

This section is concerned with extended filters that are real and, whentranslated to the origin, circularly symmetric, namely:{circumflex over (ω)}_(e)(r)=ĝ(∥r−r ₀∥),∀r∈

²,where ĝ is a function defined over

₊, and r₀=(cos θ₀, sin θ₀)∈

¹ is the focus direction. Eq. (4) shows that the beamforming function isthe 2D Fourier transform of the target filter. Leveraging the shiftproperty of Fourier transforms, we can hence express the beamformingfunction in terms of the zeroth order Hankel transform of ĝ:

$\begin{matrix}\begin{matrix}{{\omega\left( {p,\phi} \right)} = {e^{{- {j2}}\;\pi\;{{pcos}{({\phi - \theta_{0}})}}}{\int_{0}^{\infty}{\int_{0}^{2\pi}{r\;{\hat{g}(r)}e^{{- j}\; 2\pi\;{rp}\;{\cos{({\theta - \phi})}}}d\;\theta\; d\; r}}}}} \\{= {e^{{- j}\; 2\pi\;{{pcos}{({\phi - \theta_{0}})}}}\left\lbrack {2\pi{\int_{0}^{\infty}{r\;{\hat{g}(r)}{J_{0}\left( {2\pi\;{rp}} \right)}{dr}}}} \right\rbrack}} \\{{= {e^{{- j}\; 2\pi\;{{pcos}{({\phi - \theta_{0}})}}}{g(p)}}},}\end{matrix} & (5)\end{matrix}$where J₀ is the zeroth order Bessel function of the first kind. Then,g(p)=2π∫₀ ^(∞) rĝ(r)J ₀(2πrp)dr,  (6)is the Hankel transform of order zero of ĝ.

2.2.2 Beamforming in a Probabilistic Setup

Assuming that the beamforming function is such that ω∈

₁ (

²), its relationship to the target beamshape from Eq. (4) can bere-written using Eq. (5), as

$\begin{matrix}{{{\hat{\omega}(\theta)} = {{\int_{0}^{\infty}{\int_{0}^{2\pi}{\left\lbrack {e^{{- j}\; 2\pi\; p\;{co}\;{s{({\phi - \theta_{0}})}}}{g(p)}} \right\rbrack e^{j\; 2\pi\; p\;{co}\;{s{({\theta - \phi})}}}{pd}\;\phi\;{dp}}}} = {\int_{0}^{\infty}{\int_{0}^{2\pi}{{g}_{1}{\sigma_{g}(p)}e^{{- j}\; 2\pi\; p\;{co}\;{s{({\phi - \theta_{0}})}}}\mspace{14mu}\ldots\mspace{14mu} e^{j\; 2\pi\; p\;{co}\;{s{({\theta - \phi})}}}{f_{b}\left( {p,\phi} \right)}p\; d\;\phi\;{dp}}}}}},} & (7)\end{matrix}$where:∥g∥ ₁=∫₀ ^(∞) |g(p)|dpandσ_(g)(p)=sign(g(p))

Now, ƒ_(b) can be interpreted as a probability density function,corresponding to the beamforming density function introduced in sect. 1,which in polar coordinates is defined as

$\begin{matrix}{{{f_{b}\left( {p,\phi} \right)} = \frac{{g(p)}}{{g}_{1}}},{\forall{p \in {{\mathbb{R}}_{+}.}}}} & (8)\end{matrix}$

Let P: Ω→

² be a random vector with polar coordinates (P, Φ) and probabilitydensity function ƒ_(b), see eq. (8). The angular filter function is then{circumflex over (ω)}(r)=α

_(P)[σ_(g)(∥P∥)e ^(−j2π)

^(r) ⁰ ^(,P)

e ^(j2π)

^(r,P)

],∀r∈

¹,  (9)with α=∥g∥₁, r₀=(cos θ₀, sin θ₀) and

.,.

the Cartesian inner product. An expression for the extended filter{circumflex over (ω)}_(e) follows directly by extending Eq. (9) to

², that is,{circumflex over (ω)}_(e)(r)=α

_(P)[σ_(g)(∥P∥)e ^(−j2π)

^(r) ⁰ ^(,P)

e ^(j2π)

^(r,P)

],∀r∈

².  (10)

Notice that when α=1 and g≥0, this can be seen as the Fourier transformof the density function ƒ_(b), modulated by an exponential term e^(−j2π)

^(r) ⁰ ^(,P)

. Leveraging the shifting property of the Fourier transform, we finallyget{circumflex over (ω)}_(e)(r+r ₀)=

_(P)[e ^(j2π)

^(r,P)

]:=φ_(P)(r),∀r∈

²,which is the characteristic function of the random vector P.

In conclusion, in the specific case where α=1 and g≥0, the extendedfilter {circumflex over (ω)}_(e) can be seen as the characteristicfunction of the random vector P, centered on r₀, the steering direction.Similarly, the target beamshape {circumflex over (ω)} can be seen as thecharacteristic function of the random vector P, centered on r₀, andevaluated on the unit circle.

2.2.3 Empirical Beamshapes and Asymptotic Convergence

Assume that for a target beamshape {circumflex over (ω)}, withcircularly symmetric extended filter {circumflex over (ω)}_(e), wecomputed the density beamforming function ƒ_(b) as described in Section2.2.2, and sampled N independent polar coordinates from this density

$\left\{ \left( {p_{i},\phi_{i}} \right) \right\}_{{i = 1},\;\ldots\;,N}\overset{i.i.d.}{\sim}{f_{b}.}$

If we place antennas at these locations, and impose beamforming weightsω_(i) for each antenna, the resulting array will have a beamshape givenby

${{b_{N}(\theta)} = {\sum\limits_{i = 1}^{N}{w_{i}e^{j\; 2\pi\; p_{i}\;{co}\;{s{({\theta - \phi_{i}})}}}}}},{\forall{\theta \in {\left\lbrack {0,{2\pi}} \right\rbrack.}}}$

Choosing the beamforming weights as

$\begin{matrix}{{w_{i} = {\frac{\alpha}{N}{\sigma_{g}\left( p_{i} \right)}e^{{- j}\; 2\pi\; p_{i}{co}\;{s{({\phi_{i} - \theta_{0}})}}}}},{i = 1},\ldots\mspace{14mu},N,} & (11)\end{matrix}$leads to the empirical beamshape, finite-sample version of Eq. (9):

${b_{N}(\theta)} = {\frac{\alpha}{N}{\sum\limits_{i = 1}^{N}{{\sigma_{g}\left( p_{i} \right)}e^{{- j}\; 2\pi\; p_{i}{co}\;{s{({\phi_{i} - \theta_{0}})}}}{e^{j\; 2\pi\; p_{i}{co}\;{s{({\theta - \phi_{i}})}}}.}}}}$

Notice that when σ_(g)=1, the beamforming weights in Eq. (11) are simplymatched beamforming weights, which steer the beamshape around θ₀. Theempirical beamshape is an unbiased estimate of the target beamshape{circumflex over (ω)}. Moreover, depending on the function g, thisestimate can also be shown to be consistent. This follows from theuniform strong law of large numbers.

The resulting empirical beamshape writes as

${{b_{N}(\theta)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}{\psi\left( {P_{i},\theta} \right)}}}},$whereψ(P _(i),θ):=ασ_(g)(p _(i))e ^(−j2πp) ^(u) ^(cos(ϕ) ^(i) ^(-θ) ⁰ ⁾ e^(j2πp) ^(i) ^(cos(θ-ϕ) ^(i) ⁾.

Then, if the function ψ(·, θ) is continuous and measurable for all θ∈[0,2π]:

as the number of antennas N grows to infinity.

A study of the function ψ reveals that a sufficient condition for b_(N)to be a consistent estimate of {circumflex over (ω)} is for the functiong to be either strictly positive or strictly negative. For functions gthat switch signs, we can still use the classical strong law of largenumbers to show point-wise convergence. The empirical beamshapes for theLaplace, Sombrero and Gaussian functions, investigated in sections 2.3and 2.4, converge extremely quickly, approximately as 1/N.

2.3 Examples of Layout Distributions

We first discuss the traditional case, targeting one individual point,before the case of targeting whole regions. Following the framework, fora given target {circumflex over (ω)}(θ), we need to design an extendedfilter {circumflex over (ω)}_(e)(r, θ) defined on the whole plane, anddetermine an appropriate distribution ƒ_(b) for achieving the target.The derived beamforming density functions are a consequence of knownHankel transform pairs.

2.3.1 Focusing on a Single Point

To target a single direction θ₀, a simple example of filter would be{circumflex over (ω)}_(e)(r, θ)=δ(θ−θ₀)δ(r−1). The resulting beamformingfunction isω(p,ϕ)=e ^(j2πp cos(θ) ⁰ ^(-ϕ)).

Yet, as one may realize, such a function has unbounded support, and thusno beamforming density function exists. In practice, we may insteadsample over a finite disc, which is effectively achieved by an arraythat targets matched beamforming.

2.3.1.1 Sombrero Function

Consider the so-called Sombrero function,

${\hat{g}(r)} = {\sigma\;{\frac{J_{1}\left( {2\pi\;\frac{r}{\sigma}} \right)}{r}.}}$where σ>0. The beamforming density function is then, up to rescaling, auniform distribution over the disc of radius 1/σ:

${f_{b}\left( {p,\phi} \right)} \propto \left\{ {\begin{matrix}1 & {{{if}\mspace{14mu} p} \leq {1/\sigma}} \\0 & {otherwise}\end{matrix}.} \right.$

From this we can conclude that for matched beamforming, a uniformlydistributed array across a given area will get us close to the Sombrerofunction, which as the array diameter increases, converges to a Dirac.

2.3.1.2. Radial Laplace

Instead of trying to approximate a Dirac, it may be wiser to target abetter behaved function, with an explicit control on the beamshape,which decays around the focus point θ₀. This can be achieved by theradial Laplace function: ĝ(r)=e^(−π) where σ>0. The beamforming densityfunction is then, up to a rescaling function,

${f_{b}\left( {p,\phi} \right)} \propto {\frac{1}{\left( {1 + {4\pi^{2}\sigma^{2}\rho^{2}}} \right)^{\frac{3}{2}}}.}$

Example Sombrero and radial Laplace beamshapes are shown in FIGS. 4 and5. The Laplace beamshape has almost no sidelobes, with its energyfocused around the point of interest.

2.4 Targeting Regions

We now present the beamforming density functions associated to the ballindicator function and the circularly symmetric Gaussian function.

2.4.1. Ball Indicator

Consider the following disc of radius R:

${\hat{g}(r)} = \left\{ {\begin{matrix}{1,} & {r \leq R} \\{0,} & {otherwise}\end{matrix}.} \right.$

The beamforming density function is then, up to a rescaling function,

${f_{b}\left( {p,\phi} \right)} \propto {\frac{{J_{1}\left( {2\pi\;{Rp}} \right)}}{p}.}$

It is interesting to note that, due to the Hankel pair relationship,approximating a region with the ball achieves a Sombrero beamformingdensity, while approximating a Sombrero requires a uniform density in adisc.

2.4.2. Bi-Dimensional Gaussian

An alternative way to cover a region is to use a bi-dimensional Gaussianfilter:

${{\hat{g}(r)} = {\frac{1}{2\pi\;\sigma^{2}}e^{- \frac{r^{2}}{2\sigma^{2}}}}},$where σ>0. The beamforming density function is then given by, up to arescaling function

f_(b)(p, ϕ) ∝ e^(−2π²σ²p²).

Example ball indicator and Gaussian beamshapes are shown in FIGS. 6 and7. The discontinuity of the ball indicator function breaks therequirements for the convergence of the empirical beamshape (as definedabove) to be applicable. Hence, the beamshape demonstrates a significantside-lobe structure, also known as the Gibbs phenomenon. In contrast,the Gaussian is very smooth but does not isolate the portion of interestas well.

3. Examples of Applications

3.1 Wifi

We now describe an embodiment for the transmission and receipt of datausing a Wifi router. To simplify, but without limiting the scope of theinvention, we assume the router and all the devices broadcasting in theroom lie on a plane, e.g., the plane of FIG. 1.

A Wifi route with multiple antennas, carefully positioned, is used fortransmission. As input a preference function ƒ:

³→

is accessed, which identify regions of interest. To that aim, anypattern described a desired irradiation may be used, e.g., a densitymap. In variants, the preference function accessed may already be in theform of an angular function. I.e., the preference function may be theangular filter {circumflex over (ω)}(r) itself.

For example, function ƒ can indicate how each location in a living roomshould be irradiated by the router's signal based on user's preferences.In variants, the preference function may be automatically learned, usingnewer Wifi standards (802.11n, 802.11ac), whereby preferences (oraverage positions) may be automatically learned from instant locationsof devices (either passively detected or with help of explicit statuscommunication).

Consider for example a room as in FIG. 1 (approximately 10.5×7 m). TheWifi router 10 is located in the upper right corner. The darker regionsof the overlaid density plot correspond to locations where users aremost likely to use their electronic devices, and hence where they needgood power coverage from the router.

Based on the preference function accessed, an optimal layout is obtainedand, subsequently, an optimal beamshape is achieved, which covers wellthe preferential areas, while compensating for signal attenuation whentravelling in the area of interest.

Preferred embodiments would approximate the preference function in termsof Gaussian functions, so as to enable analytical calculation of thebeamforming function.

Note that most embodiments evoked herein assume the devices to lie inthe far field. Indeed, no beamforming would be necessary if the deviceswere in the near field. This assumption is believed to hold for allreasonable modes of operation. For example for a frequency of operationof ƒ₀=2.4 GHz (as typical for Wifi signals), and for a single antenna oflength l=15 cm, every device located at a distance from the antennagreater than d₀=2l²/λ≈3.6 mm can be considered as lying in the farfield.

An example of The Wifi router 10 is depicted in FIG. 8, which iscomposed of 27 antennas and has a radius of 25 cm. The positions {p₁, .. . , p₂₇}∈

² of the individual antennas {A₁, . . . , A₂₇} are restricted to

² for simplicity and the center of the router is denoted by p₀∈

². Said positions are determined according to principles describedearlier. In FIG. 8, a few

antennas positions are explicitly spotted, i.e., the projections of afew antenna positions are represented on a plane parallel to the averageplane of the router 10, the average plane assumed to coincide with theplane of interest.

An example radial filter {circumflex over (ω)}(θ) is shown in FIGS. 2Aand 2B. It is unwrapped in FIG. 3 and plotted over the interval [π/2,π], by seeing it as a function of θ∈[−π, π], i.e., the polar coordinateof the vector r on the circle. To facilitate the design of such afilter, we may approximate it as a contraction of three Gaussianfunctions (as otherwise depicted in FIG. 3). The maxima of {circumflexover (ω)}(θ) typically correspond to maxima of the projections on theunit circle. More generally, all the parameters involved may be refinedfor {circumflex over (ω)}(θ) to fit the projections.

We can compute the beamforming function ω(p) by taking the(well-defined) Fourier transform of {circumflex over (ω)}_(e)(r). Thefunction {circumflex over (ω)}_(e) is designed so as to extrapolate{circumflex over (ω)} to the plane assuming one has, radially,essentially the same characteristic extension as tangentially. Yet,other approaches would convene as well, provided that the extendedfilter {circumflex over (ω)}_(e) is “thicker” than {circumflex over(ω)}. As a general rule though, {circumflex over (ω)}_(e) may suitablyextend {circumflex over (ω)} radially by assuming the radial deviationequal to the tangential deviation. Note that in the above example,{circumflex over (ω)}_(e) is not exactly equal to {circumflex over (ω)}on the circle. In variants, one may, however, constrain it to takeexactly the same values as {circumflex over (ω)} on the circle, whilegiving it a given thickness on each side of the circle, as in the caseof the ball indicator function.

In general, a characteristic radial extension of {circumflex over(ω)}_(e) will delimit a circular region with radius less than 1 (i.e.,think of a ball indicator function of radius less than 1, the value ofunit-circle radius). However, in exceptional cases (large regions), thecharacteristic radial extension of {circumflex over (ω)}_(e) may delimita circular region with radius larger than 1, but always less than 2(i.e., the critical radius for which the entire unit-circle would becontained in the extension. Hence, larger radii (although theoreticallyconceivable) would not provide any substantial benefit. For example,when using the ball indicator function, a radius of 2 or greater wouldalways lead to the same beamshape, filtering uniformly across allpossible directions.

${{b(r)} = {\frac{1}{\beta}{\sum\limits_{i = 1}^{L}{{\omega\left( \frac{p_{i}}{\lambda} \right)}e^{j\; 2\pi\;{\langle{r,\frac{p_{i}}{\lambda}}\rangle}}}}}},$

Assuming L antennas in the router. β is the normalization factor, i.e.,

${\beta = {\sum\limits_{i = 1}^{L}{{\omega\left( \frac{p_{i}}{\lambda} \right)}}^{2}}},$to avoid noise magnification due to beamforming. Because the beamshapeas defined above is an aliased version of the target angular filter,small side-lobes may result in the radiation pattern (see FIGS. 4-7).Yet, thanks to antenna layouts as obtained in embodiments, such lobeswill remain small, if not negligible, and the resultant radiationpattern will cover well the regions of interest.

3.2 Mobile Phone

Here we briefly describe embodiments for beamforming using a collectionof 3G/4G (or in the future 5G) transmitters. The goal is to determinewhat beamshape should be used at each transmitter. For a giventransmitter of a set, a preferred embodiment is as follows:

-   -   Compute the angular filter by taking the radial projection of        the preference function, which amounts to circular cuts of the        global density function defined over the whole region;    -   Approximate this angular filter by a sum of weighted Gaussian        functions, see, e.g., FIG. 3;    -   Extend this filter to the plane, using a technique as described        earlier, and compute its Fourier transform analytically, to        obtain the density function ƒ_(b);    -   Sample positions of antennas composing the transmitter and        position the antennas at the determined positions (note that the        position of the transmitters too may be optimized); and    -   Compute the weights to be applied to each antenna composing the        transmitter by sampling the Fourier transform.

For example, one may wish to beamform so as to cover optimally a part ofa city, based on a priori knowledge of the probable positions ofclients.

The beamshapes obtained according to the above procedure approximatevery well the associated angular filter. With this technique, thenetwork's users will have a better signal's reception, as less power isdissipated in unnecessary areas of the city. This is a huge advantagecompared to current technologies in use, where broadcasting is isotropicand independent from the device distribution.

3.3 Ultrasound

Ultrasound consists of transmit beamforming (beamforming to target theultrasound output at a certain point) and receive beamforming(beamforming the resultant echoes). Two concurrent operations are thusneeded, one of the transmit and one of the receive type. The transmittedand received signals are linked in that case because the echoes areshaped by the transmit beamforming.

Ultrasound phased arrays are often 1D (with all transducers aligned on asingle axis, as assumed in FIG. 9), generating 2D images. More recently,2D arrays of transducers have been proposed, which generate 3D images.In all cases, application of the present invention can be made toimprove the transducer layouts.

The potential benefits of present embodiments for ultrasound include:

-   -   Faster generation of images;    -   More accurate images;    -   Resolution can more carefully be adjusted;    -   Easier design of systems for 3D imaging;    -   More accurate videos (due to faster update potential); and    -   A lower number of transducers is necessary, enabling low-cost        and portable devices.

The general (known) method for transmit beamforming consists of inducingdelays (a delay law) between the pulses emitted by each of thetransducers. The goal is to achieve a narrow focus a certain distanceaway for a desired angle. This is equivalent to performing a focusedbeamforming for a given angle each time. The echoes from the resultantbeam are then received by each transducer. They are then combined by aweighted sum to produce a single image, using the receive beam. Thefinal image can be obtained by scanning across many angles. In this modeof operation, the whole image is desired, and focused beamforming isjust a means to obtain it. Each sweep has high SNR at the point offocus, and reduced SNR around it, with side-lobes also distorting theresultant image.

Embodiments of the invention modify this known method by allowing forvariation in the delays and magnitudes at each transducer, thus inducingphase and magnitude changes in the resultant beamforming.

3.4 Radio Astronomy

Modern radio telescope correlate the signals measured by thousands ofantennas at various location on the ground, in order to infer an imageof the sky. To reduce the amount of data sent to the central processorand increase the signal to noise ratio, the antennas are groupedtogether in stations and beamformed via matched beamforming. As we haveseen previously, matched beamforming maximizes the power coming from onedirection in the sky, but sees little about the rest of the sky. Hence,surveying large portions of the sky with matched beamforming requiresmultiple observations, where matched beamforming is steered towardsvarious locations on the sky in order to cover a whole portion ofinterest. This is suboptimal and time consuming, and one would benefitmuch more from a beamshape which could capture the power of the signalscoming from a wider range of directions.

Thus, an example of application to radio astronomy, for a givencollection of antennas that have a modifiable layout, is:

-   -   A desired spatial filter is given on a unit sphere.    -   This desired spatial filter is extended to the 3D space.    -   An optimal layout of antennas is derived and the antennas are        correspondingly located; and    -   The subsequent beamforming function is obtained, which is        sampled at the optimized locations of the antennas to obtain the        weights.

These weights are then applied to beamform at each time-instance.

4. Technical Implementation Details

4.2 Computerized Units and Apparatuses

Computerized systems and devices can be suitably designed forimplementing embodiments of the present invention as described herein.In that respect, it can be appreciated that the methods described hereinare largely non-interactive and automated. In exemplary embodiments, themethods described herein can be implemented either in an interactive,partly-interactive or non-interactive system. The methods describedherein can be implemented in software (e.g., firmware), hardware, or acombination thereof. In exemplary embodiments, the methods describedherein are implemented in software, as an executable program, the latterexecuted by suitable digital processing devices. More generally,embodiments of the present invention can be implemented whereingeneral-purpose digital computers, such as personal computers,workstations, etc., are used.

For instance, the system 100 depicted in FIG. 11 schematicallyrepresents a computerized unit 101, e.g., a general- or specific-purposecomputer, which may be used as part of any of the apparatuses 10, 10 orsystems 1 described earlier. As such, the unit 101 may interact withreceivers, transmitters and/or transceivers, e.g., via converters andI/O units 145-155.

In exemplary embodiments, in terms of hardware architecture, as shown inFIG. 11, the unit 101 includes a processor 105, memory 110 coupled to amemory controller 115. One or more input and/or output (I/O) devices145, 150, 155 (or peripherals) are communicatively coupled via a localinput/output controller 135. The input/output controller 135 can be orinclude, but is not limited to, one or more buses and a system bus 140,as is known in the art. The input/output controller 135 may haveadditional elements, which are omitted for simplicity, such ascontrollers, buffers (caches), drivers, repeaters, and receivers, toenable communications. Further, the local interface may include address,control, and/or data connections to enable appropriate communicationsamong the aforementioned components.

The processor 105 is a hardware device for executing software,particularly that stored in memory 110. The processor 105 can be anycustom made or commercially available processor, a central processingunit (CPU), an auxiliary processor among several processors associatedwith the computer 101, a semiconductor based microprocessor (in the formof a microchip or chip set), or generally any device for executingsoftware instructions.

The memory 110 can include any one or combination of volatile memoryelements (e.g., random access memory) and nonvolatile memory elements.Moreover, the memory 110 may incorporate electronic, magnetic, optical,and/or other types of storage media. Note that the memory 110 can have adistributed architecture, where various components are situated remotefrom one another, but can be accessed by the processor 105.

The software in memory 110 may include one or more separate programs,each of which comprises an ordered listing of executable instructionsfor implementing logical functions. In the example of FIG. 11, thesoftware in the memory 110 includes methods described herein inaccordance with exemplary embodiments and, in particular, a suitableoperating system (OS) 111. The OS 111 essentially controls the executionof other computer programs and provides scheduling, input-outputcontrol, file and data management, memory management, and communicationcontrol and related services.

The methods described herein may be in the form of a source program,executable program (object code), script, or any other entity comprisinga set of instructions to be performed. When in a source program form,then the program needs to be translated via a compiler, assembler,interpreter, or the like, as known per se, which may or may not beincluded within the memory 110, so as to operate properly in connectionwith the OS 111. Furthermore, the methods can be written as an objectoriented programming language, which has classes of data and methods, ora procedure programming language, which has routines, subroutines,and/or functions.

Possibly, a conventional keyboard and mouse can be coupled to theinput/output controller 135. Other I/O devices 140-155 may include or beconnected to other hardware devices 10, as noted earlier.

In addition, the I/O devices 140-155 may further include or be connectedto devices 10 that communicate both inputs and outputs. The system 100can further include a display controller 125 coupled to a display 130.In exemplary embodiments, the system 100 can further include a networkinterface or transceiver 160 for coupling to a network 165, to enable,in turn, data communication to/from other, external components.

The network 165 transmits and receives data between the unit 101 andexternal systems. The network 165 is possibly implemented in a wirelessfashion, e.g., using wireless protocols and technologies, such as Wifi,WiMax, etc. The network 165 may be a fixed wireless network, a wirelesslocal area network (LAN), a wireless wide area network (WAN) a personalarea network (PAN), a virtual private network (VPN), intranet or othersuitable network system and includes equipment for receiving andtransmitting signals.

The network 165 can also be an IP-based network for communicationbetween the unit 101 and any external server, client and the like via abroadband connection. In exemplary embodiments, network 165 can be amanaged IP network administered by a service provider. Besides, thenetwork 165 can be a packet-switched network such as a LAN, WAN,Internet network, etc.

If the unit 101 is a PC, workstation, intelligent device or the like,the software in the memory 110 may further include a basic input outputsystem (BIOS). The BIOS is stored in ROM so that the BIOS can beexecuted when the computer 101 is activated.

When the unit 101 is in operation, the processor 105 is configured toexecute software stored within the memory 110, to communicate data toand from the memory 110, and to generally control operations of thecomputer 101 pursuant to the software. The methods described herein andthe OS 111, in whole or in part are read by the processor 105, typicallybuffered within the processor 105, and then executed. When the methodsdescribed herein are implemented in software, the methods can be storedon any computer readable medium, such as storage 120, for use by or inconnection with any computer related system or method.

4.2 Computer Program Product

The present invention may be an apparatus, a method, and/or a computerprogram product. The computer program product may include a computerreadable storage medium (or media) having computer readable programinstructions thereon for causing a processor to carry out aspects of thepresent invention.

The computer readable storage medium can be a tangible device that canretain and store instructions for use by an instruction executiondevice. The computer readable storage medium may be, for example, but isnot limited to, an electronic storage device, a magnetic storage device,an optical storage device, an electromagnetic storage device, asemiconductor storage device, or any suitable combination of theforegoing. A non-exhaustive list of more specific examples of thecomputer readable storage medium includes the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a static random access memory (SRAM), a portablecompact disc read-only memory (CD-ROM), a digital versatile disk (DVD),a memory stick, a floppy disk, a mechanically encoded device such aspunch-cards or raised structures in a groove having instructionsrecorded thereon, and any suitable combination of the foregoing. Acomputer readable storage medium, as used herein, is not to be construedas being transitory signals per se, such as radio waves or other freelypropagating electromagnetic waves, electromagnetic waves propagatingthrough a waveguide or other transmission media (e.g., light pulsespassing through a fiber-optic cable), or electrical signals transmittedthrough a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network may comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

Computer readable program instructions for carrying out operations ofthe present invention may be assembler instructions,instruction-set-architecture (ISA) instructions, machine instructions,machine dependent instructions, microcode, firmware instructions,state-setting data, or either source code or object code written in anycombination of one or more programming languages, including an objectoriented programming language such as Smalltalk, C++ or the like, andconventional procedural programming languages, such as the C programminglanguage or similar programming languages. The computer readable programinstructions may execute entirely on the user's computer, partly on theuser's computer, as a stand-alone software package, partly on the user'scomputer and partly on a remote computer or entirely on the remotecomputer or server. In the latter scenario, the remote computer may beconnected to the user's computer through any type of network, includinga local area network (LAN) or a wide area network (WAN), or theconnection may be made to an external computer (for example, through theInternet using an Internet Service Provider). In some embodiments,electronic circuitry including, for example, programmable logiccircuitry, field-programmable gate arrays (FPGA), or programmable logicarrays (PLA) may execute the computer readable program instructions byutilizing state information of the computer readable programinstructions to personalize the electronic circuitry, in order toperform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

These computer readable program instructions may be provided to aprocessor of a general purpose computer, special purpose computer, orother programmable data processing apparatus to produce a machine, suchthat the instructions, which execute via the processor of the computeror other programmable data processing apparatus, create means forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks. These computer readable program instructionsmay also be stored in a computer readable storage medium that can directa computer, a programmable data processing apparatus, and/or otherdevices to function in a particular manner, such that the computerreadable storage medium having instructions stored therein comprises anarticle of manufacture including instructions which implement aspects ofthe function/act specified in the flowchart and/or block diagram blockor blocks.

The computer readable program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational steps to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the block may occur out of theorder noted in the figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

While the present invention has been described with reference to alimited number of embodiments, variants and the accompanying drawings,it will be understood by those skilled in the art that various changesmay be made and equivalents may be substituted without departing fromthe scope of the present invention. In particular, a feature(device-like or method-like) recited in a given embodiment, variant orshown in a drawing may be combined with or replace another feature inanother embodiment, variant or drawing, without departing from the scopeof the present invention. Various combinations of the features describedin respect of any of the above embodiments or variants may accordinglybe contemplated, that remain within the scope of the appended claims. Inaddition, many minor modifications may be made to adapt a particularsituation or material to the teachings of the present invention withoutdeparting from its scope. Therefore, it is intended that the presentinvention not be limited to the particular embodiments disclosed, butthat the present invention will include all embodiments falling withinthe scope of the appended claims. In addition, many other variants thanexplicitly touched above can be contemplated. For example, otherapplications than those explicitly mentioned may benefit frombeamforming methods as described herein.

What is claimed is:
 1. A system comprising an apparatus with a set of Ntransducers {A_(i)}_(i=1, . . . , N), wherein: the transducers areconfigured, in the apparatus, for receiving wave signals from and/ortransmitting wave signals to one or more regions{R_(m)}_(m=1, . . . , M) of interest in an n-dimensional space, with n=2or 3; and the transducers are arranged at positions{p_(i)}_(i=i, . . . , N) within said n-dimensional space, wherein saidpositions {p_(i)}_(i=i, . . . , N) corresponds to arguments of values{ƒ_(b)(p_(i))}_(i=1, . . . , N) of a density function ƒ_(b)(p) obtainedbased on a Fourier transform ω(p) of a n-dimensional spatial filterfunction {circumflex over (ω)}_(e)(r), the latter determined so as tomatch projections {P_(m)}_(m=1, . . . , M) of the one or more regions{R_(m)}_(m=1, . . . , M) of interest onto an n−1-dimensional spherecentered on the apparatus.
 2. The system according to claim 1, wherein:the transducers are movably mounted in the apparatus; and the system isfurther configured to: determine an n-dimensional spatial filterfunction {circumflex over (ω)}_(e)(r), which matches projections{P_(m)}_(m=1, . . . , M) of one or more regions {R_(m)}_(m=1, . . . , M)of interest onto an n−1-dimensional sphere centered on the apparatus;obtain a density function ƒ_(b)(p), based on a Fourier transform ω(p) ofthe determined spatial filter function {circumflex over (ω)}_(e) (r);determine, within said n-dimensional space, positions{p_(i)}_(i=1, . . . , N) for each of the N transducers, based on theobtained density function ƒ_(b) (p); and position the N transducersaccording to the positions {p_(i)}_(i=1, . . . , N) determined.
 3. Thesystem according to claim 1, further comprising: one or more targetslocated in said one or more regions {R_(m)}_(m=1, . . . , M) ofinterest, respectively, wherein the transducers are configured forreceiving wave signals from and/or transmitting wave signals to said oneor more targets.
 4. The system according to claim 1, wherein: saidapparatus is a transmission apparatus and said transducers form a set oftransmitters {A_(i)}_(i=1, . . . , N), the latter configured fortransmitting wave signals to said one or more regions{R_(m)}_(m=1, . . . , M) of interest, and wherein the apparatus isfurther configured for beamforming a wave signal for said set oftransmitters {A_(i)}_(i=1, . . . , N).
 5. The system according to claim1, wherein: said apparatus is a receiving apparatus and said transducersform a set of receivers {A_(i)}_(i=1, . . . , N), the latter configuredfor receiving wave signals from said one or more regions{R_(m)}_(m=1, . . . , M) of interest, and wherein the apparatus isfurther configured for computing a beamformed signal from said set ofreceivers {A_(i)}_(i=1, . . . , N).
 6. The system according to claim 1,wherein: said apparatus is a transceiver apparatus and said transducersform a set of transceivers {A_(i)}_(i=1, . . . , N), the latterconfigured for receiving wave signals from and transmitting wave signalsto said one or more regions {R_(m)}_(m=1, . . . , M) of interest, andwherein the apparatus is further configured for both computing abeamformed signal from and beamforming a wave signal for said set oftransmitters {A_(i)}_(i=1, . . . , N).
 7. A non-transitory computerreadable medium comprising computer executable instructions which whenexecuted by a computer cause the computer to perform a method fordetermining positions {p_(i)}_(i=1, . . . , N) of transducers{A_(i)}_(i=1, . . . , N) of an apparatus, wherein the transducers areconfigured for receiving wave signals from and/or transmitting wavesignals to one or more regions {R_(m)}_(m=1, . . . , M) of interest inan n-dimensional space, with n=2 or 3, the method comprising the stepsof: determine a n-dimensional spatial filter function {circumflex over(ω)}_(e) (r), which matches projections {P_(m)}_(m=1, . . . , M) of theone or more regions {R_(m)}_(m=1, . . . , M) of interest onto ann−1-dimensional sphere centered on the apparatus; obtain a densityfunction ƒ_(b)(p), based on a Fourier transform ω(p) of the determinedspatial filter function {circumflex over (ω)}_(e)(r); and determine,within said n-dimensional space, a position p_(i) for each of Ntransducers, based on the obtained density function ƒ_(b)(p) and aprescribed number N of transducers.